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ANALYSING AMINO ACIDS IN HUMAN GALANIN AND ITS

RECEPTORS - GRAPH THEORETICAL APPROACH

Suresh Singh G.a and Akhil C. K.b

a

Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram – 695581, Kerala, India,

b

Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram – 695581, Kerala, India,

Acknoldgements

We wish to thank Dr Achuthsankar S. Nair, Head of the Department, Department of Computational Biology and Bioinformatics, University of Kerala, Thiruvananthapuram along with the faculty members, Vijayalakshmi B. and Biji C. L. for their fruitful suggestions and constant encouragements.

Abstract: Graph theoretical analysis is an important area of research in biological networks. In this work we define some new graphs called bipartite Pt-graphs and their physicochemical subgraphs for peptides/proteins and their receptors based on the physicochemical properties of amino acids. Here we analyze bipartite Pt-graphs and their physicochemical subPt-graphs of human galanin and its three receptors graph theoretically. From the graph theoretical analysis of bipartite Pt-graphs and the physicochemical subPt-graphs we get some observations about the relations among the amino acids, physicochemical properties, galanin and its receptors. By a graph theoretical parameter of physicochemical subgraphs we get all the collections of maximum independent pairs of amino acids which connect the galanin and receptors by sharing exactly 𝑛 (𝑛 = 1,2,3, … ) common physicochemical properties. These analyses can be used to study all the relationships between peptide/protein ligands and their receptors and this may help in the field of drug designing.

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1. Introduction

Proteins are polymers of amino acids, with each amino acid residue joined to its neighbour by a specific type of covalent bond [3]. Twenty different types of amino acids are commonly found in peptide/protein. The sequence of amino acids in a protein is characteristic of that protein and is called its primary structure [3]. Peptides/proteins are the compounds of amino acids in which a carboxyl group of one is united with an amino group of another. Neuropeptides are peptides formed and released by neurons. They are involved in a wide range of brain functions.

Galanin is a neuropeptide of 30 amino acids in humans and 29 amino acids in other species [4]. It is expressed in a wide range of tissues including the brain, spinal cord and gut. Its signaling occurs through three G protein-coupled receptors. It is linked to a number of diseases including Alzheimer’s disease, epilepsy, depression, eating disorders, cancer, etc.

In [7], we can see so many graph theoretical applications in various fields. Amino acid network with in protein was studied by S. Kundu [5]. By using some physicochemical properties (Hydrophobicity, Hydrophilicity, Polarity, Non-polarity, Aliphaticity, Aromaticity and Charge (Positive and Negative)) of amino acids, the amino acid network was studied by Adil Akthar and Nisha Gohan graph theoretically [1]. The centralities in amino acid networks were used by Adil Akhtar and Tazid Ali [2]. By using the concept of amino acid network we defined and analysed the peptide/protein graph (Pt-graph) and species peptide/protein graph (SPt-graph) of galanin present in fourteen species of animals graph theoretically [6]. In this work we define and analyse new graphs - bipartite Pt-graphs and physicochemical subgraphs (physicochemical property-wise) - of human galanin and its three receptors on the basis of the physicochemical properties of amino acids. The maximum matching of physicochemical subgraphs is done to get all the collections of maximum independent pairs of amino acids which connect the galanin and its receptors by sharing exactly 𝑛 (𝑛 = 1,2,3, … ) common physicochemical properties of amino acids. This method can be applied for all relationships between peptides/proteins and their receptors and this may help in the field of drug designing.

2. Basic Concepts of Graph Theory

Definition 2.1: A Graph [7] 𝒒 is a pair 𝒒 = (𝒱, β„°) consisting of a finite set 𝒱 and a set β„° of 2-element subsets of 𝒱. The elements of 𝒱 are called vertices and elements of β„° are called edges. The set 𝒱 is known as the vertex set of 𝒒 and β„° as the edge set of 𝒒. Two vertices 𝑒 and 𝑣 of 𝒒 are said to be adjacent, if an edge join 𝑒 and 𝑣, and two edges are adjacent if they have common vertex. The number of vertices in a graph 𝒒 is called its order and the number of edges is its size. A graph with 𝑝 vertices and π‘ž edges is said to be a (𝑝, π‘ž) graph.

Definition 2.2: Centrality measures in Graphs [2] are the vertex representation which gives the relative importance within the graph. A centrality is a real-valued function 𝑓 which assigns every vertex 𝑣 ∈ 𝒱 of a given graph 𝒒 a value 𝑓(𝑣) ∈ ℝ.

Definition 2.3: Let 𝒒 be an arbitrary (𝑝, π‘ž) graph. β„³ βŠ‚ β„°(𝒒) is said to be a matching in 𝒒 if its

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be saturated by a matching β„³ (β„³- saturated) or matched vertex with respect to β„³ if 𝑣 is incident with an edge of β„³. Otherwise, the vertex is said to be unsaturated by β„³ (β„³- unsaturated) or a single vertex with respect to β„³. A matching β„³ of a graph 𝒒 is said to be a perfect matching if all the vertices of 𝒒 are saturated by β„³.

Definition 2.4: A Pt-graph is defined as a graph 𝒒 = (𝒱, β„°) of a peptide/protein in which the vertex set, 𝒱 is the collection of all different amino acids presented in the peptide/protein and weight of a vertex in 𝒒 is the number of times it appears in the sequence of the peptide/protein. Two vertices are said to be adjacent in 𝒒 if they are consecutive elements in the sequence and also have at least one common physicochemical property.

For all terminologies and notations not mentioned in this work, we follow [7] (related to graph theory) and [3] (related to biology).

Remark: Weight of a vertex implies the frequency of occurrence of a specific amino acid in a sequence. Obviously greater the weight of a vertex of a Pt-graph implies greater the characteristics of those particular amino acid can be attributed to the peptide/protein. Also the centrality measures of a Pt-graph help us to determine the number of amino acids possess inter-relationships with each other.

3. Bipartite Pt-graphs and physicochemical subgraphs of human galanin and its receptors

In this section we define some new graphs called bipartite Pt-graphs and physicochemical subgraphs for peptides/proteins and its receptors. Also we construct and analyze bipartite Pt-graphs and their physicochemical subPt-graphs of human galanin and its receptors using π’ž-sets of corresponding Pt-graphs.

efinition 3.1: A bipartite Pt-graphis defined as a simple bipartite graph 𝒒 = (𝒱, β„°) of a peptide/protein and its receptors with 𝒳 and 𝒴 as the partitions of the vertex sets of the corresponding Pt-graphs of the peptide/protein and its receptors respectively. Two vertices π‘₯ ∈ 𝒳 and 𝑦 ∈ 𝒴 are said to be adjacent if they have atleast one common physicochemical property.

Definition 3.2: A π’ž-set of a Pt-graph of a peptide/protein is defined as the subset of the vertex set whose elements are the amino acids which recieve the highest centrality measures for each physicochemical properties of amino acids.

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Figure 1: Pt-graph of human galanin

Figure 2: Pt-graph of receptor-1

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Figure 4: Pt-graph of receptor-3

From Pt-graphs we get the π’ž-set from the highest centralities of amino acids for each physicochemical property. For human galanin, 𝐺5 (Hydrophoic and Non-polar), 𝑆4 and 𝑁3 (Hydrophilic and polar), 𝐿4 (Aliphatic), π‘Œ1 and π‘Š1 (Aromatic), 𝐻2, 𝐾1 and 𝑅1 (Positive), 𝐷1

(Negative) are the amino acids which receive the highest centralities. Hence the π’ž-set for the galanin is {𝐺5, 𝑆4, 𝑁3, 𝐿4, π‘Œ1, π‘Š1, 𝐻2, 𝐾1, 𝑅1, 𝐷1}. Similarly we get the π’ž-sets for the Pt-graphs of receptor-1, receptor-2 and receptor-3. Then the π’ž-set for receptor-1 is {𝐴29, 𝑁14, 𝑆32, 𝐿40, 𝐹25, 𝐾20, 𝐸10}, π’ž-set for receptor-2 is {𝐴46, 𝑆28, 𝑉30, 𝐹17, 𝑅27, 𝐷8, 𝐼18, π‘Š8} and π’ž -set for receptor-3 is {𝐴65, 𝐺32, 𝑃25, 𝑆19, 𝐿50, 𝐹8, 𝑅39, 𝐸8, 𝐷9, 𝑉28}.

Next we analyse the bipartite Pt-graphs of human galanin and its receptors.

Bipartite Pt-graphs of human galanin and its receptors

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Figure 6: Bipartite Pt-graph of galanin and receptor-2

Figure7: Bipartite Pt-graph of galanin and receptor-3

Definition 3.3: Physicochemical subgraphs β„‹π‘˜π‘–(for 𝑖 = 1,2,3, … ) of a bipartite Pt-graph 𝒒 of a peptide/protein and π‘˜ receptors is defined as a subgraph whose vertex sets are same as that of 𝒒 and two vertices in the different partitions are adjacent if they have exactly 𝑖 common physicochemical properties.

Remark: Let 𝒒 = (𝒱, β„°) be a bipartite Pt-graph of a peptide/protein and its π‘˜ receptors with 𝒳and 𝒴 as the partitions of the vertex sets and let β„‹π‘˜π‘– 𝒳, 𝒴𝑖 , where 𝒴𝑖 βŠ† 𝒴 (for 𝑖 = 1,2, … , 𝑛) be 𝑛 physicochemical subgraphs ,

Then,

𝒴𝑖 𝑖=1,2,3,…𝑛

= βˆ… and 𝒴𝑖

𝑖=1,2,3,…𝑛

= 𝒴

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Physicochemical subgraphs of galanin and receptor-1

Figure 8: β„‹11subgraph

Figure 9: β„‹12subgraph

Figure 10: β„‹13subgraph

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Physicochemical subgraphs of galanin and receptor-2

Figure 12: β„‹21subgraph

Figure 13:β„‹22subgraph

Figure 14: β„‹23subgraph

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Physicochemical subgraphs of galanin and receptor-3

Figure 16: β„‹31subgraph

Figure 17:β„‹32subgraph

Figure 18: β„‹33subgraph

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Next we obtain the maximum matching of β„‹π‘˜π‘– subgraphs of bipartite Pt-graphs of human galanin and its receptors. Table 1 represents all the collections of maximum independent pairs of amino acids which connecting the human galanin and its receptors by sharing exactly 𝑖 (𝑖 = 1,2,3,4) common physicochemical properties.

Receptors π’Œ

Maximum matching of π‘―π’Œπ’Š subgraphs

π“—π’ŒπŸ subgraphs π“—π’ŒπŸ subgraphs π“—π’ŒπŸ‘ subgraphs π“—π’ŒπŸ’ subgraphs

π’Œ = 𝟏

(𝐺5, 𝐾20), 𝑆4, 𝐿40 , 𝐿4, 𝑆32 , π‘Œ1, 𝐸10 ,

𝐻2, 𝐹25 , (𝐾1, 𝐴29)

(𝐺5, 𝑁14), 𝑆4, 𝐴29 , π‘Œ1, 𝐿40 , π‘Š1, 𝑆32 ,

(𝐻2, 𝐸10)

(𝐺5, 𝐿40), 𝐿4, 𝐴29 , π‘Š1, 𝐹25 , 𝐻2, 𝐾20 ,

𝐾1, 𝑁14 , 𝑅1, 𝑆32 ,

𝑁3, 𝐸10

(𝐺5, 𝐴29), 𝑆4, 𝑁14 , 𝐿4, 𝐿40 , 𝐾1, 𝐾20 ,

𝐷1, 𝐸10 , 𝑁3, 𝑆32

π’Œ = 𝟐

(𝐺5, 𝐷8), 𝑆4, 𝑉30 , 𝐿4, 𝑆28 , π‘Š1, 𝑅27 ,

𝐻2, 𝐹17 , 𝐾1, π‘Š8 ,

𝐷1, 𝐴46 , (𝑁3, 𝐼18)

𝑆4, 𝐴46 , (𝐿4, π‘Š8), π‘Œ1, 𝐼18 , π‘Š1, 𝑉30 ,

(𝐻2, 𝑆28)

(𝐺5, 𝐼18), 𝐿4, 𝐴46 , π‘Š1, 𝐹17 , 𝐻2, 𝑅27 ,

𝐾1, 𝑆28 , 𝑅1, 𝐷8

(𝐺5, 𝐴46), 𝐿4, 𝐼18 , π‘Œ1, π‘Š8 , 𝐾1, 𝑅27 ,

𝐷1, 𝐷8 , 𝑁3, 𝑆28

π’Œ = πŸ‘

(𝐺5, 𝐷9), 𝑆4, 𝐿50 , 𝐿4, 𝑆19 , π‘Œ1, 𝑅39 ,

π‘Š1, 𝐸8 , 𝐻2, 𝐹15 ,

𝐾1, 𝐴65 , 𝑅1, 𝐺32 ,

𝐷1, 𝑃25 , 𝑁3, 𝑉28

𝐺5, 𝑆19 , (𝑆4, 𝐴65), π‘Œ1, 𝑉28 , π‘Š1, 𝐿50 ,

𝐻2, 𝐸8 , (𝑁3, 𝑃25)

(𝐺5, 𝐿50), 𝑆4, 𝑉28 , 𝐿4, 𝐴65 , π‘Š1, 𝐹15 ,

𝐻2, 𝑅39 , 𝐾1, 𝑆19 ,

𝑅1, 𝐸8 , 𝑁3, 𝐷9

(𝐺5, 𝐴65), 𝐿4, 𝑉28 , 𝐾1, 𝑅39 , 𝐷1, 𝐸8 ,

𝑁3, 𝑆19

Table 1: Maximum matching of π“—π’Œπ’Šsubgraphs of human galanin and its receptors

Let π’’π‘˜ 𝒳, π’΄π‘˜ (for receptors π‘˜ = 1,2,3) be three bipartite Pt-graphs of human galanin and its receptors, where 𝒳 is the vertex set of Pt-graph of human galanin and 𝒴1, 𝒴2 and 𝒴3 are the vertex sets of Pt-graphs of receptor-1, receptor-2 and receptor-3 respectively. Also let β„‹π‘˜π‘– 𝒳, π’΄π‘˜ be the physicochemical subgraphs of π’’π‘˜ 𝒳, π’΄π‘˜ , where 𝑖 indicates the number of common physicochemical properties of amino acids. Also let we denote the amino acids with physicochemical properties as

𝑃1+= Hydrophobic, 𝑃1βˆ’= Hydrophilic, 𝑃2+=Polar, 𝑃2βˆ’= Non-polar,

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By analysing β„‹π‘˜π‘– subgraphs, we get some observations about the physicochmical property-wise connections of amino acids of galanin and its receptors.

Observation 3.1: In the analysis of the bipartite Pt-graphs of human galanin and receptors, we obtain 𝑁14, 𝑆32 (in receptor 1), 𝑆28, π‘Š8 (in receptor 2) and 𝑆19 (in receptor 3) are the amino acids which receiving all highest centralities. Also we obtain 𝐺5, 𝑆4, π‘Œ1, π‘Š1 and 𝑁3 are the common amino acids of human galanin which receive the highest centralities in all the bipartite Pt-graphs.

Observation 3.2: In the physicochemical subgraphs with amino acids sharing exactly one common property, the connections of amino acids of galanin to receptor-1, receptor-2 and receptor-3 are

(1) 𝑃1+ is not connected with 𝑃1+ and 𝑃1βˆ’ is not connected with 𝑃1βˆ’ (2) 𝑃2βˆ’ is not connected with 𝑃2βˆ’.

(3) 𝑃3+ is not connected with both 𝑃3+ and 𝑃3βˆ’ but 𝑃3βˆ’ is not connected with 𝑃3+. (4) 𝑃4+ and 𝑃4βˆ’ are not connected with both 𝑃4+ and 𝑃4βˆ’.

Observation 3.3: In the physicochemical subgraphs with amino acids sharing exactly two common properties, the connections of amino acids of galanin to receptor-1 and receptor-3 are

(1) 𝑃1+ and 𝑃1βˆ’ are connected with both 𝑃1+ and 𝑃1βˆ’ . (2) 𝑃2βˆ’ is not connected with 𝑃2βˆ’.

(3) 𝑃3+ is not connected with, 𝑃3+, 𝑃3βˆ’ and 𝑃30 𝑃3βˆ’ is not connected with 𝑃3βˆ’

𝑃30 is not connected with 𝑃3+ and 𝑃3βˆ’. (4) 𝑃4+ is not connected with 𝑃4βˆ’

𝑃4βˆ’ is not connected with 𝑃4+, 𝑃4βˆ’ and 𝑃40 𝑃40 is not connected with 𝑃4+ and 𝑃4βˆ’.

Observation 3.4: In the physicochemical subgraphs with amino acids sharing exactly two common properties, the connections of amino acids of galanin to receptor-2 are

(1) 𝑃1+ and 𝑃1βˆ’ are conected with both 𝑃1+ and 𝑃1βˆ’. (2) 𝑃2βˆ’ is not connected with 𝑃2βˆ’.

(3) 𝑃3+ is not connected with 𝑃3+ and 𝑃30 𝑃3βˆ’ is not connected with 𝑃3βˆ’ 𝑃30 is not connected with 𝑃3+. (4) 𝑃4+ is not connected with 𝑃4βˆ’

𝑃4βˆ’ is not connected with 𝑃4+, 𝑃4βˆ’ and 𝑃40 𝑃40 is not connected with 𝑃4+ and 𝑃4βˆ’.

Remark: In the physicochemical subgraphs with amino acids sharing exactly two common properties, the amino acids 𝑃3+ and 𝑃30 of galanin are conncted with the amino acids 𝑃3βˆ’ of receptor-2

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Observation 3.5: In the physicochemical subgraphs with amino acids sharing exactly three common properties, the connections of amino acids of galanin to receptor-1, receptor-2 and receptor-3 are

(1) 𝑃1+ is not connected with 𝑃1βˆ’ and 𝑃1βˆ’ is not connected with 𝑃1+. (2) 𝑃2βˆ’ is not connected with 𝑃2+.

(3) 𝑃3+ and 𝑃3βˆ’ are not connected with 𝑃3+. (4) 𝑃4+ is not connected with 𝑃4+.

Observation 3.6: In the physicochemical subgraphs with amino acids sharing exactly four common properties, the connections of amino acids of galanin to receptor-1, receptor-2 and receptor-3 are

(1) 𝑃1βˆ’ have more neighbours than 𝑃1+ of 𝑋. (2) 𝑃2βˆ’ have more neighbours than 𝑃2+ of 𝑋. (3) 𝑃4βˆ’ have more neighbours than 𝑃4+ of 𝑋.

Observation 3.7: There is no neighbours for 𝑃3βˆ’ in receptor-2 of the physicochemical subgraph with amino acids sharing exactly four common properties.

Observation 3.8: The physicochemical subgraph of galanin and receptor-3 with amino acids sharing exactly one common property (figure 16) is the subgraph having a maximum matching which is the only perfect matching among all the physicochemical subgraphs of galanin and its receptors.

Conclusion

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sharing exactly one common property (figure 16) is the subgraph having a maximum matching which which is the only perfect matching among all the physicochemical subgraphs of galanin and its receptors. These analyses can be used to study all the relationships between peptide/protein ligands and their receptors and this may help in the field of drug designing.

References

[1] Adil Akhtar and Nisha Gohain, Graph theoretic approach to analyze amino acid network, Int. J. Adv. Appl. Math. And Mech. 2(3) (2015) 31-37.

[2] Adil Akhtar and Tazid Ali, Analysis of Unweighted Amino Acids Network, Hindawi Publishing Corporation, Vol. 2014, Article ID 350276 (2014) 6 pages.

[3] David L. Nelson and Michael M. Cox, Lehninger Principles of Biochemistry, W. H Freeman and Company (2008).

[4] Mitsukawa K., Lu X., Bartfai T., Galanin, Galanin receptors and drug targets. Cell. Mol. Life. Sci. 65, 1976-180510.1007/s00018-008-8153-8 (2008)

[5] S. Kundu, Amino acid network with in protein, Physica A, 346 (2005) 104-109.

[6] Suresh Singh G., Akhil C. K., Analysing Amino Acids in Galanin – Graph Theoretical Appraoch, International Journal on Recent and Innovation Trends in Computing and Communication (IJRITCC), June 17 Volume 5 Issue 6, ISSN: 2321-8169, PP: 342-346 (2017).

[7] Suresh Singh G., Graph Theory, PHI Learning Private Limited (2010).

Figure

Figure 2: Pt-graph of receptor-1
Figure 4: Pt-graph of receptor-3
Figure 6: Bipartite Pt-graph of galanin and receptor-2
Figure 8: β„‹1
+4

References

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